EMPIRICAL DISTRIBUTION FUNCTION

In statistics, an 'empirical distribution function' is a cumulative probability distribution function that concentrates probability 1/''n'' at each of the ''n'' numbers in a sample.
Let $X\_1,ldots,X\_n$ be iid random variables in $mathbb\{R\}$ with the cdf ''F(x)''.
The empirical distribution function
$F\_n(x)$ based on sample $X\_1,ldots,X\_n$ is a step function defined by
:
rac{1}{n} sum_{i=1}^n I(X_i le x),
where ''I''(''A'') is the indicator of event ''A''.
For fixed ''x'', $I(X\_ileq\; x)$ is a Bernoulli random variable with parameter ''p=F(x)'', hence $nF\_n(x)$ is a binomial random variable with mean ''nF(x)'' and variance ''nF(x)(1-F(x))''.

Contents

Asymptotical properties

See also

Asymptotical properties

★ By the strong law of large numbers,
:: $F\_n(x)\; o\; F(x)$ almost surely for fixed ''x''.
:In other words, $F\_n(x)$ is consistent unbiased estimator of the cumulative distribution function ''F(x)''.

★ By the central limit theorem,
:: $sqrt\{n\}(F\_n(x)-F(x))$ converges in distribution to a normal distribution ''N(0,F(x)(1-F(x)))'' for fixed ''x''.
:The Berry–Esséen theorem provides the rate of this convergence.

★ By the Glivenko-Cantelli theorem $F\_n(x)\; o\; F(x)$ uniformly over ''x'', that is
:: $|F\_n(x)-F(x)|\_infty\; o\; 0$ with probability 1.
:The Dvoretzky-Kiefer-Wolfowitz inequality provides the rate of this convergence.

★ Kolmogorov showed that
:: $sqrt\{n\}|F\_n(x)-F(x)|\_infty$ converges in distribution to the Kolmogorov distribution, provided that ''F(x)'' is continuous.
:The Kolmogorov-Smirnov test for ''goodness-of-fit'' is based on this fact.

★ By Donsker's theorem,
:: $sqrt\{n\}(F\_n-F)$, as a process indexed by ''x'', converges weakly in $ell^infty(mathbb\{R\})$ to a Brownian bridge ''B(F(x))''.

See also

★ Càdlàg functions

★ Empirical probability

★ Empirical_process